The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

g(f(x, y), z) → f(x, g(y, z))
g(h(x, y), z) → g(x, f(y, z))
g(x, h(y, z)) → h(g(x, y), z)

Rewrite Strategy: INNERMOST

(1) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1]
transitions:
f0(0, 0) → 0
h0(0, 0) → 0
g0(0, 0) → 1
g1(0, 0) → 2
f1(0, 2) → 1
f1(0, 0) → 3
g1(0, 3) → 1
g1(0, 0) → 4
h1(4, 0) → 1
g1(0, 3) → 2
f1(0, 2) → 2
f1(0, 2) → 4
f1(0, 3) → 3
g1(0, 3) → 4
h1(4, 0) → 2
h1(4, 0) → 4

(2) BOUNDS(1, n^1)